Contents

Idea

The Freyd cover of a category is a special case of Artin gluing. Given a category 𝒯\mathcal{T} and a functorF:𝒯→SetF: \mathcal{T} \to Set, the Artin gluing of FF is the comma categorySet↓FSet \downarrow F whose objects are triples (X,ξ,U)(X, \xi, U) where:

XX is a set

UU is an object of 𝒯\mathcal{T}

ξ\xi is a function X→F(U)X \to F(U).

The Freyd cover is the special case F=𝒯(1,−)F = \mathcal{T} (1, -).

The Freyd cover is sometimes known as the Sierpinski cone or scone, because in topos theory it behaves similarly to the cone on a space, but with the interval[0,1][0,1] replaced by the Sierpinski space.

Properties

Relation to the initial topos

One of the first applications of the Freyd cover was to deduce facts about the initial topos? (initial with respect to logical morphisms — also called the free topos). They were originally proved by syntactic means; the conceptual proofs of the lemma and theorem below are due to Freyd.

Lemma

For any category CC with a terminal object 1\mathbf{1}, the terminal object of the Freyd cover C^\widehat{C} is connected and projective, i.e., the representable Γ=C^(1,−):C^→Set\Gamma = \widehat{C}(1, -) \colon \widehat{C} \to Set preserves any colimits that exist.

Proof

To check that Γop:C^op→Setop\Gamma^{op} \colon \widehat{C}^{op} \to Set^{op} preserves limits, it suffices to check that the composite

preserves limits, because the contravariant power set functor P=2−P = 2^- is monadic. But it is easily checked that this composite is the contravariant representable given by (2,1,2→Γ(1))(2, \mathbf{1}, 2 \to \Gamma(\mathbf{1})).

Theorem

Proof

We divide the argument into three segments:

The hom-functor preserves finite limits, so by general properties of Artin gluing, the Freyd cover 𝒯^\widehat{\mathcal{T}} is also a topos. Observe that 𝒯\mathcal{T} is equivalent to the slice 𝒯^/M\widehat{\mathcal{T}}/M where MM is the object (∅,1,∅→Γ(1))(\emptyset, \mathbf{1}, \emptyset \to \Gamma(\mathbf{1})). Since pulling back to a slice is a logical functor, we have a logical functor

π:𝒯^→𝒯\pi \colon \widehat{\mathcal{T}} \to \mathcal{T}

Since 𝒯\mathcal{T} is initial, π\pi is a retraction for the unique logical functor i:𝒯→𝒯^i \colon \mathcal{T} \to \widehat{\mathcal{T}}.

We have maps 𝒯(1,−)→𝒯^(i1,i−)≅𝒯^(1,i−)\mathcal{T}(1, -) \to \widehat{\mathcal{T}}(i 1, i-) \cong \widehat{\mathcal{T}}(1, i-) (the isomorphism comes from i1≅1i 1 \cong 1, which is clear since ii is logical), and 𝒯^(1,i−)→𝒯(π1,πi−)≅𝒯(1,−)\widehat{\mathcal{T}}(1, i-) \to \mathcal{T}(\pi 1, \pi i-) \cong \mathcal{T}(1, -) since π\pi is logical and retracts ii. Their composite must be the identity on 𝒯(1,−)\mathcal{T}(1, -), because there is only one such endomorphism, using the Yoneda lemma and terminality of 11.

Finally, since 𝒯(1,−)\mathcal{T}(1, -) is a retract of a functor 𝒯^(1,i−)\widehat{\mathcal{T}}(1, i-) that preserves finite colimits (by the lemma, and the fact that the logical functor ii preserves finite colimits), it must also preserve finite colimits.

This is important because it implies that the internal logic of the free topos (which is exactly “intuitionistic higher-order logic”) satisfies the following properties:

The disjunction property: if “P or Q” is provable in the empty context, then either P is so provable, or Q is so provable. (Note that this clearly fails in the presence of excluded middle.)

The existence property: if “there exists an x∈Ax\in A such that P(x)P(x)” is provable in the empty context, then there exists a global elementx:1→Ax\colon 1\to A such that P(x)P(x) is provable in the empty context. (Again, this is clearly a constructivity property.)

The negation property: False is not provable in the empty context.

As a local topos

The Freyd cover of a topos is a local topos, and in fact freely so. Every local topos is a retract of a Freyd cover.